Singular

D.6.11.6 invariants

Usage:

invariants(INPUT); INPUT list or poly

Assume:

INPUT is the output of develop(f), or of extdevelop(develop(f),n), or one entry of the list of HN data computed by hnexpansion(f[,"ess"]).

Return:

list INV of the following format:

INV[1]: intvec (characteristic exponents) INV[2]: intvec (generators of the semigroup) INV[3]: intvec (Puiseux pairs, 1st components) INV[4]: intvec (Puiseux pairs, 2nd components) INV[5]: int (degree of the conductor) INV[6]: intvec (sequence of multiplicities)

If INPUT contains no valid HN expansion, the empty list is returned.

Assume:

INPUT is a bivariate polynomial f, or the output of hnexpansion(f), or the list of HN data computed by hnexpansion(f [,"ess"]).

Return:

list INV, such that INV[i] coincides with the output of invariants(develop(f[i])), where f[i] is the i-th branch of f, and the last entry of INV contains further invariants of f in the format:

INV[last][1] : intmat (contact matrix of the branches) INV[last][2] : intmat (intersection multiplicities of the branches) INV[last][3] : int (delta invariant of f)

Note:

In case the Hamburger-Noether expansion of the curve f is needed for other purposes as well it is better to calculate this first with the aid of hnexpansion and use it as input instead of the polynomial itself.

LIB "hnoether.lib";
ring exring=0,(x,y),dp;
list Hne=develop(y4+2x3y2+x6+x5y);
list INV=invariants(Hne);
INV[1];                   // the characteristic exponents
==> 4,6,7
INV[2];                   // the generators of the semigroup of values
==> 4,6,13
INV[3],INV[4];            // the Puiseux pairs in packed form
==> 3,7 2,2
INV[5] div 2;             // the delta-invariant
==> 8
INV[6];                   // the sequence of multiplicities
==> 4,2,2,1,1
// To display the invariants more 'nicely':
displayInvariants(Hne);
==>  characteristic exponents  : 4,6,7
==>  generators of semigroup   : 4,6,13
==>  Puiseux pairs             : (3,2)(7,2)
==>  degree of the conductor   : 16
==>  delta invariant           : 8
==>  sequence of multiplicities: 4,2,2,1,1
/////////////////////////////
INV=invariants((x2-y3)*(x3-y5));
INV[1][1];                // the characteristic exponents of the first branch
==> 2,3
INV[2][6];                // the sequence of multiplicities of the second branch
==> 3,2,1,1
print(INV[size(INV)][1]);         // the contact matrix of the branches
==>      0     3
==>      3     0
print(INV[size(INV)][2]);         // the intersection numbers of the branches
==>      0     9
==>      9     0
INV[size(INV)][3];                // the delta invariant of the curve
==> 14